The importance of the knowledge of the fundamental properties of stochastic systems and processes has been recently acknowledged by a growing portion of the scientific and engineering community. Among other properties of stochastic processes, the nature of the noise component which contaminates the pure signal of the system is of major importance. The term “noisy signal” or “raw signal” whenever referred to in this application, refers to a signal which comprises a noise component and a pure signal which are inseparable. Throughout this application, the term “noise” refers to any random or unknown component whose exact behavior cannot be exactly predicted, but knowing its probability density function is highly valuable. Also, the term “variance” relates to the second moment of the probability density function and is used as is common in the art of Statistics and Probability theories. Moreover, throughout this application the terms “machine”, “system” and “process” are used interchangeably with respect to the method of the invention. An accurate estimation of the noise properties can provide to the system designer very important tools for improving the system behavior. An accurate determination of the noise properties is particularly important for dynamical systems where non-linear behavior is expected and in which the noise may seriously alter any estimation of the states of the system, if not to cause a total divergence of the parameters of the system model. Such conditions are particularly common in non-linear systems when modeled by recursive or adaptive methods such as Weiner or Kalman filtering. The principles and theory of Kalman and Weiner filtering are described, for example, in Gelb, A., “Applied Optimal Estimation”, Chapter 1, pp. 1-7, The MIT Press, Cambridge, Mass., 1974.
The following United States patents are believed to represent the state of the art for Signal estimation, noise characteristics, and Kalman and adaptive filtering in applicable systems: U.S. Pat. Nos. 6,829,534; 6,740,518; 6,718,259; 6,658,261; 6,836,679; 6,754,293; and 6,697,492.
The theory of non-linear filtering and its applications are discussed in:    (a) Grewal, M. S. et al., Kalman Filtering, Prentice-Hall, 1993;    (b) Jazwinski, A. H., Stochastic Processes and Filtering Theory, Academic Press, New York, 1970, chapters 1 and 2, pp. 1-13;    (c) Gelb, A., Applied Optimal Estimation, The MIT Press, Cambridge, Mass., 1974 Chapter 1, pp. 1-7; and    (d) Wiener, N., Journal of Mathematical and Physical Sciences 2, 132 (1923).
The art of signal processing, probability and stochastic processes and noise characteristics are also discussed in:    (a) Bruno Aiazzi et al., IEEE Signal Processing Lett. 6 138 (1999);    (b) R. Chandramouli et al., “Probability, Random Variables and Stochastic Processes”, A. Papoulis, McGraw-Hill USA, (1965);    (c) IEEE Signal Processing Lett. 6 129;    (d) Zbyszek P. Karkuszewski, Christopher Jarzynski, and Wojciech H. Zurek, Phys Rev. Lett. 89, 170405 (2002);    (e) A. F. Faruqi and K. J. Turner Applied Mathematics and Computation, 115, 213 (2000);′    (f) J. P. M. Heald and J. Stark, Phys. Rev. Lett. 84, 2366 (2000);    (g) A. A. Dorogovtsev, Stochastic Analysis and Random Maps in Hilbert Space, VSP Publishing, The Netherlands, (1994) (in particular see the consideration for high-order stochastic derivative in chap. 1);    (h) H. Kleinert and S. V. Shabanov, Phys. Lett. A, 235, 105, (1997);    (i) Elachi, C., Science, 209, 1073-1082, (1980);    (j) Valeri Kontorovich et al., IEEE Signal Processing Lett. 3, 19 (1996);    (k) Steve Kay., IEEE Signal Processing Lett. 5, 318 (1998);    (l) Michael I. Tribelsky, Phys. Rev. Lett. 89, 070201 (2002).
The theory of curve fitting, differentiation and high order derivatives is discussed in:    (a) G. Di Nunno, Pure Mathematics 12, 1, (2001); and    (b) K. Weierstrass, Mathematische Werke, Bd. III, Berlin 1903, pp. 1-17.
It is an object of the present invention to provide a method for the statistical separation and determination of the noise properties from the noisy signal.
It is another object of the invention to provide such a method that can be performed in real-time.
It is still another object of the present invention to provide such a method for characterizing the noise which is adaptive.
It is still another object of the present invention to provide said method for characterizing the noise that can determine not only the variance of the noisy signal, but also the type of the probability density function (pd) of the noise component.
It is still another object of the present invention to provide said method for characterizing the noise that does not depend on a priori knowledge of the structure of the pure signal.
It is still another object of the present invention to provide said method for characterizing the noise that does not depend on the structure of the pure signal.
It is still another object of the present invention to provide said method for characterizing the noise that involves defining a window of the analyzed signal, and given said window, the method does not depend on any accumulative information outside said window boundaries.
Other objects and advantages of the present invention will become apparent as the description proceeds.